Coco: Corecursion with Compositional Heterogeneous Productivity

Abstract: Contemporary proof assistants impose restrictive syntactic guardedness conditions that reject many valid corecursive definitions. Existing approaches to overcome these restrictions present a fundamental trade-off between coverage and automation. We present Compositional Heterogeneous Productivity (CHP), a theoretical framework that unifies high automation with extensive coverage for corecursive definitions. CHP introduces heterogeneous productivity applicable to functions with diverse domain and codomain types, including non-coinductive types. Its key innovation is compositionality: the productivity of composite functions is systematically computed from their components, enabling modular reasoning about complex corecursive patterns. Building on CHP, we develop Coco, a corecursion library for Rocq that provides extensive automation for productivity computation and fixed-point generation.